Aerosol particle properties can be derived by mea-surement of a variety of scattering properties.These properties describe the particle’s influence onthe Earth’s radiation budget, on clouds, and on pre-cipitation, as well as their role in chemical processesof the troposphere and the stratosphere. This mayinclude extinction or scattering information at mul-tiple wavelengths, scattering information at multipleangles, or multiple-scattering information. Herethe inversion of particle properties from lidar mea-surements of backscatter and extinction at multiplewavelengths is discussed.
The inversion requires the solution of a Fredholmsystem of integral equations of the first kind. Themathematical model consists of two integral equations
The author [email protected]! is with the Instituteof Mathematics, University of Potsdam, PF 601553, D-14415 Pots-dam, Germany.
Received 4 April 2000; revised manuscript received 31 August2000.
for the backscatter and extinction coefficients b andaaer:
baer~l, z! 5 *rmin
rmax
kp~r, l; m!n~r, z!dr
5 *rmin
rmax
pr2Qp~r, l; m!n~r, z!dr, (1)
aaer~l, z! 5 *rmin
rmax
kext~r, l; m!n~r, z!dr
5 *rmin
rmax
pr2Qext~r, l; m!n~r, z!dr. (2)
Here r denotes the particle radius, m is the refractiveindex, rmin and rmax represent suitable lower and up-per limits, respectively, of realistic radii, l is the wave-length, z is the height, n is the aerosol size distributionwe are looking for, kp is the backscatter, and kext is theextinction kernel. The kernel functions reflect theshape, size, and material composition of the particles.We assume Mie particles. When actual lidar data are
20 March 2001 y Vol. 40, No. 9 y APPLIED OPTICS 1329
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3,4
e
Table 1. Different Common Setup Cases of Multispectral Lidar Devices
1
used, the refractive index is also an unknown. There-fore the problem is a nonlinear ill-posed problem, i.e.,the correct kernel function is also unknown. On theother hand, all mathematical algorithms currently inuse assume the kernel function to be known correctly,with the following formulas1 providing the extinctionand the backscatter efficiencies Qp and Qext of Eqs. ~1!and ~2!:
Qp 51
k2r2U(
n51
`
~2n 1 1!~21!n~an 2 cn!U2
,
Qext 52
k2r2 (n51
`
~2n 1 1!Re~an 1 cn!, (3)
here k is the wave number defined by k 5 2pyl andn and cn are the coefficients that one gets from the
boundary conditions. The standard formula1 forcomputing Qp and Qext in the case of homogeneousspheres is
where jn and yn are spherical Bessel functions andv 5 kr.
The lidar setup2 of the Institute for Troposphericesearch in Leipzig, Germany, typically uses sixackscatter and two Raman channels; see the firstetup case in Table 1. This number cannot, how-ver, be achieved with standard currently availableidar systems, which work with only three backscat-er and one Raman channel; see the fifth setup case inable 1. Moreover, it is probable that during a mea-urement campaign one or two channels may fail.his inspired the consideration of the other threeetup cases, which are shown as cases 2–4 in Table 1.he resulting determination of the aerosol size dis-
ribution function n~r! from only a small number ofackscatter and extinction measurements by meansf a system of Fredholm integral equations of the firstind is the most familiar and common example notnly for a compact operator but also for a severelyll-posed inverse problem.
In the past few years different inversion techniquesave been proposed for solving similar multiwave-
ength ill-posed inversions of more special operators.
330 APPLIED OPTICS y Vol. 40, No. 9 y 20 March 2001
The algorithms are based on quadrature discreti-zations that contain no regularization effect; this willbe considered in Section 2. The first algorithm3
deals with the inversion of optical thickness data bymeans of the Tikhonov regularization ~TR! at sevenqually spaced wavelengths. The second algorithm4
deals with scattering intensity data by means of trun-cated singular-value decomposition ~TSVD! regular-ization by use of between five and eight wavelengthsat a scattering angle of 176°. References 5 and 6propose a review of different regularization tech-niques and compare some of these ~constraint leastsquares, TSVD, Landweber iterations7! by determi-nation of the global aerosol distribution, for whichsolar radiation data with known basic aerosol com-ponents are used. Other interesting works in re-lated fields describe, for example, channel selectionby use of optical depth data,8 a purely graphic tech-nique for parameter estimations of known distribu-tion shapes,9 or the aerosol mass concentrationdistribution.10 Also discussed are analytic inver-sions11,12 of the special nonabsorbing anomalous dif-fraction extinction operator12 or the optical thicknessdata inversion by truncated analytic eigenfunctionexpansion,11 with the Mie theory extinction kernelused as a product kernel Qext~r, l; m! 5 Qext~kr; m!.An overview and a critical review of different inver-sion techniques for in situ devices can be found in Ref.13. References 14–20 contain the most related al-gorithms with respect to lidar ill-posed inversion.The methods of Refs. 14–17 work with projectiondiscretization by means of first-order B splines andby means of TR16 at eight wavelengths or a Bayesian-based iterative regularization method,15 respectively.References 18–20 work with second-order B splinesby means of TR but still with eight wavelengths. Ofthese, only Ref. 20 treats the case of an unknownrefractive index.
In this paper a new hybrid regularization methodis proposed that uses variable projection dimensionand variable B-spline order as well as TSVD simul-taneously for that severely ill-posed inversion. Fi-nally, the method can handle the inversion with onlyone extinction and three backscatter coefficients up to10% noise and for two extinction and six backscattercoefficients up to 20% noise. In addition, the un-known refractive index can be captured. The paperis organized as follows. After the mathematicalbackground is described in Section 2, the speciallydeveloped hybrid regularization method is proposedin Section 3 and the projection method as a regular-ization tool is discussed. Moreover, inversion re-sults for simulated data are shown in order to find asuitable B-spline basis. In Section 4 several mea-surement situations with simulated noiseless andnoisy data for numerous different distributions withknown refractive indices are studied, as well as sev-eral examples with additional unknown refractive in-dices. In these latter examples we are able toretrieve simultaneously both the distribution and therefractive index. Finally, conclusions are given inSection 5.
fo
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I
aK
st
p
i
wnhIK
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ai
satl
2. Mathematical Background
Equations such as Eqs. ~1! and ~2!, i.e., with compactoperators K, are always ill-posed, generally on allthree counts ~existence, uniqueness, and stability!,or which stability21 means a solution that changesnly slightly with a slight change in the problem.We consider an equation of the form Kx 5 y, where
K : X 3 Y is a compact, linear ~but not necessarilyelf-adjoint! operator from a Hilbert space X into ailbert space Y. The general theory of compact op-
rators evolved from the theory of integral operatorsf the form
y~l! 5 *rmin
rmax
k~l, r!x~r!dr. (6)
ndeed, if k~ z , z ! is a square integrable over @lmin,lmax# 3 @rmin, rmax#, then it is a well-known classicresult that K is a compact operator from L2@rmin, rmax#into L2@lmin, lmax#. In addition, if k~ z , z ! is contin-uous, as in our case, then K is a compact operatorfrom C@rmin, rmax# into C@lmin, lmax#. The operatorsK*K : X3X and KK* : Y3Y are compact self-adjointlinear operators, where K* is the adjoint operator ofK. The nonzero eigenvalues of K*K or of KK* ~theyhave the same eigenvalues! can be enumerated asl1 $ l2 $ . . . . If we designate by v1, v2, . . . , anssociated sequence of orthonormal eigenvectors of*K, then $vj% is complete in the range R~K*K! 5
N~K!' @orthogonal complement of the null space N~K!of K#. Let mj 5 =lj; then Kvj 5 mjuj and K*uj 5 mjvj.Moreover, KK*uj 5 mjKvj 5 mj
2uj 5 ljuj, and it is notdifficult to see that the orthonormal eigenvectors $uj%of KK* form a complete orthogonal set for R~KK*! 5N~K*!'. The system $vj, uj; mj% is called a singularsystem for K and the numbers mj are called singularvalues of K. In the nondegenerate infinite-dimensional case the values li~i 5 1, 2, . . . ! of theingular-value expansion ~SVE! of the operator clus-er at zero. Any function of the form
x 5 (j51
` ^y, uj&
mjvj 1 w (7)
with w [ N~K! is a solution. In general, we deal withnoisy data yd and often hold yd ¸ R~K!. Then we lookfor the set of all least-squares solutions. This set isclosed and convex. Therefore a unique least-squares solution K†y with smallest norm exists. Themapping K† is the Moore–Penrose generalized
seudoinverse of K. K† is unbounded, i.e., K† is dis-continuous, and it holds
K†yd 5 (j51
` ^yd, uj&
mjvj. (8)
This representation of K†yd shows clearly that K† isunbounded if R~K! is infinite dimensional. Indeed, aperturbation in y of the form dun gives a new right-hand side in Kx 5 y of the form yd 5 y 1 dun satisfyingiy 2 ydi 5 d. Yet the generalized solution satisfiesK†y 2 K†ydi 5 dymn3 ` as n3 `. Slight changes
in y~l! @see Eq. ~6!# can provide large changes in thesolution x~r!.
Now we try to substitute the ill-posed problem witha nearby well-posed problem. There are differentpossibilities of making problems stable. First,sometimes it is possible to use objective a priori in-formation about the solution as one is able to reducethe set of permissible solutions to a compact set.Such a suitable reduction to D , X of the operator Kcan help to overcome the instability. This is thedescriptive regularization22; see also the remark inSection 5. Second, additional information about thenoise d, i.e., the measurement errors, may be helpful,e.g., deterministic or stochastic information about thenoise. Third, one can deal with subjective a prioriinformation by using a support functional
V : D , X3 R, V~x! 5 min!, x [ D , X, (9)
here x is compatible with the data. This is theondescriptive regularization for which the solutionas certain desired features, such as smoothness.n general, regularizations are families of operatorsg with g [ R†.23
Kg : Y3 X with limg30Kg y 5 K†y for all y [ D~K†!,i.e., the convergence is pointwise on D~K†!. The pa-rameter g is the regularization parameter. In thecase of noisy data yd with iy 2 ydi # d, we determinea solution xg
d 5 Kg yd. However, the total errorconsists of two parts, i.e., two summands:
xgd 2 x 5 Kg~yd 2 y! 1 ~Kg 2 K†!y. (10)
he first part is the data error, and the second part ishe approximation or regularization error. If g3 0
the approximation error tends to zero whereas thedata error tends to infinity. Therefore the total er-ror can never be zero. We are looking for a trade-offbetween these two terms, i.e., for an optimal regular-ization parameter g that minimizes the total error.Many regularization methods can be found in Ref. 24.
First, the most popular and well-known regulariza-tion is the classic TR or the iterative TR.25 Therere other examples such as truncated and general-zed truncated singular-value decomposition23,26
~SVD!, asymptotic regularization,24 iterative meth-ods ~e.g., linear Landweber iterations,7 nonlinearconjugate gradient iterations,27,28 Lardy method orSchulz method25!, mollifier methods ~i.e., first smooththe function y by mollification29–31 and then approx-imate the mollified function!, or maximum entropymethods.32
Second, there are also many different discretiza-tion possibilities: simple classic quadrature meth-ods with different weights, collocation points andnodes, degenerate kernel approximations ~expan-ions by eigenfunctions or by orthonormal systems,pproximations by Taylor or interpolation!, projec-ion methods ~Galerkin, moment, collocation, oreast-squares methods33–35!, and multilevel algo-
rithms.36 The first two discretization methods ap-proximate the underlying equation, whereas theprojection methods approximate the solution. On
20 March 2001 y Vol. 40, No. 9 y APPLIED OPTICS 1331
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o
38,39
hc
s
1
the one hand, it is now possible to combine any reg-ularization method with any discretization methodwhen one recognizes that each numerical algorithmhas its advantages and its drawbacks. On the otherhand, it is well known that the straightforward naivequadrature approach to integral equations of the firstkind is usually unsatisfactory and leads to disastrousresults if the data contain any errors or uncertainties.The approximations become even worse for finerquadrature discretization schemes. Fortunately, ina projection method, discretization and regulariza-tion work hand in hand to produce the linear system;this happens because pure projection into finite-dimensional spaces itself acts as regularization inwhich the regularization parameter is the dimensionn, i.e., g 5 1yn ~see Section 3!. This is in sharpcontrast to the first method, in which the discretiza-tion by quadrature takes place first without any reg-ularization and then an additional regularization isperformed after the discretization is complete.37
Therefore the application of the quadrature methodsis, in principle, confined to equations of the secondkind, and these are generally well-posed.
Because my experience shows that such a standardmethod works well for only small measurement er-rors I decided to develop a hybrid regularizationmethod to amalgamate different advantages of vari-ous methods.
3. Hybrid Regularization Method
Based on knowledge of the properties of different stan-dard discretization and regularization techniques, aspecially developed hybrid regularization technique isproposed that can handle different lidar systems thatwork with various values and numbers of wavelengthsand up to 20% noisy data. The regularization processconsists of a discretization part by means of projectionand a regularization part; the first part itself has reg-ularization properties. To ensure that both regular-ization stages work hand in hand we use threeregularization parameters simultaneously. The firstis the projection dimension; the second is the order ofthe B-spline basis used; and the last, which is impor-tant during the solution of the resulting system oflinear equations, is the level of the TSVD.
A. Regularization by Discretization by Means ofProjection Methods
Let X be a normed space and U , X be a nontrivialubspace. A bounded linear operator P : X3 U with
the property Pt 5 t for all t [ U is called a projectionperator from X onto U. Two important examples for
projection operators are given by the orthogonal pro-jection, e.g., Galerkin methods, and by the interpola-tion projection, e.g., collocation methods. We studythe last one. Let Xn , X and Ym , Y be two subspaceswith dim Xn 5 n, dim Ym 5 m, and with basis
332 APPLIED OPTICS y Vol. 40, No. 9 y 20 March 2001
Different possibilities of taking out a projectionexist. The dual projection method is preferred, andthe approximation xn [ X with
xn 5 b1f1 1 · · · 1 bnfn (12)
is the solution of the projected equation
Pm Kxn 5 Pm y. (13)
Here, Pm : Y 3 Ym is an appropriate projection op-erator; let Km :5 PmK. The projected equationyields a system of linear equations for all n unknowncoordinates bi, i 5 1, . . . , n. The error of the pro-jection method depends on how well the exact solu-tion can be approximated by elements of the subspaceXn. Increasing n and m will make the error smallerin relation to the discretization. However, becauseof ill-conditioning, the computation will be more con-taminated by errors of the given data y. Note that inactual numerical computations such errors will occurautomatically because of round-off effects on thedata. On the other hand, if n and m are small, thenthe approximation is robust against errors in y butwill be inaccurate because of a large discretizationerror. This problem is the same as in Eq. ~10!, whichcalls for a compromise in the choice of discretizationparameters n and m. In general, all discretizations
ave shown that they lead to matrices with a highondition number,40 which grows with the increasing
dimension of the matrix. Besides, the degree of ill-conditioning of a matrix is measured by the conditionnumber, which is the ratio between the largest andthe smallest singular value. Let K†y [ X be theolution of Kx 5 y and m 5 n. By xn [ Xn, we denote
the unique solution of the equation PnKxn 5 Pny.We can represent the solutions xn [ Xn in the formxn 5 Rny, where Rn : Y3 Xn , X is defined by Rn :5~PnKuXn
!†Pn : Y3 Xn , X. The projection method iscalled convergent if the approximate solutions xn [Xn converge to the exact solution K†y [ X for everyy [ R~K!, i.e., if
Rn y 5 ~Pn KuXn!†Pn y3K†y, n3 `, (14)
for every y [ D~K†!. We observe that this definitionof convergence coincides with the definition of theprevious section, i.e., a regularization strategy for theequation Kx 5 y with regularization parameter g~d!5 1yn. Therefore the projection method converges ifand only if Rn is a regularization strategy for theequation Kx 5 y. Hence pure discretization turnsout to be a regularization method. For convergenceproofs of the special case m 5 n see Refs. 24 and 33.In more detail, we will now see that there is a hiddenregularization parameter, namely, the smallest sin-gular value nn of the operator Kn :5 PnK, where Pn isthe orthogonal projector onto Yn with iPni 5 1. Fora stability analysis we assume that noisy data yd are
d †
1
d
ths
eU
Ss
d
kFp
available with iPn~y 2 y !i # d. Because iKn i 5ynn we obtain
ixnd 2 x†i # ixn 2 x†i 1 iKn
†Pn~y 2 yd!i
# ixn 2 x†i 1 iKn†id
# ixn 2 x†i 1d
nn. (15)
First, because limn3`nn 5 0, we must choose asmaller n for larger errors. Second, if Yn has a fixed
imension n, we can ask how to choose Yn to maxi-mize nn. This would be an optimal choice in thesense that we obtain a minimal error with respect tothe noise for fixed n. For compact operators, as inour case, we can give an answer to this question.Let K be compact with the singular system $vj, uj; mj%;hen nn # mn holds ~see Ref. 26!. Obviously equalityolds for the special choice of Yn 5 Un :5pan$u1, . . . , un%. The resulting method is the trun-
cated SVE. This choice is also optimal with respectto the approximation error term. In this sense, thechoices of the basis functions fj [ Xn and rj [ Yn areessential in Eq. ~11!.
B. Collocation Method with Variable Order of B Splines
A collocation projection is proposed because y isknown at only a few measurement points m and,additionally, a reduction in the run time on a com-puter is desired. In a collocation method Eq. ~13! isquivalent to ~Kxn!~li! 5 y~li! for all i 5 1, . . . , m.sing Eq. ~11! to find a solution of the form xn 5 ¥j51
n
bjfj leads to the finite linear system
(j51
n
~Kfj!~li! bj 5 y~li!, i 5 1, . . . , m. (16)
The application to the integral equations of the formof Eq. ~6! results in Ab 5 y~li!, with
Aij :5 *rmin
rmax
k~li, r!fj~r!dr. (17)
In our case it will be useful to study several differentbasis functions fi
~ j!, i 5 1, . . . , nj. Thus fi~1!, i 5
1, . . . , n1, can span a space Xn1, whereas fi
~2!, i 51, . . . , n2, can span Xn2
. This matter is clarified inection 4. Numerous experimental calculationsuggest that, in general, Galerkin41 and collocation
matrices generated by basis functions of small sup-port, e.g., B splines, are better conditioned and thecomputations are easier than those arising from amore classic basis with global support, e.g., Legendreand Chebyshev polynomials. Moreover, by reason ofrelation ~15!, the basis functions should be near thesingular functions of the operator.
We get a well-conditioned basis of the B splines B 5$N1k, . . . , Nnk% with the recursion formulas ~x@ti,ti11#
where t1 # . . . # ts are the extended nodes42 andNik~r! are the B splines of the order of k and i 51, . . . , n. The support of Nik is a local one; it is Nik~r!$ 0 for all r [ R. Nik~r! is a piecewise polynomial of
egree g # k 2 1 and the Nik, i 5 1, . . . , n are locallinear independent. See Fig. 1~a! for an example of
5 4 and n 5 9. Three points are important to note.irst, because a distribution function has to be zero atoints r smaller than rmin and at points larger than
rmax it is proposed to set b1 5 0 and bn 5 0. Second,in the total B-spline basis case @see Fig. 1~a!#, at-tempting to recover the distribution often leads to theGibbs phenomenon, i.e., high-frequency oscillationsat the base of the original distribution. These oscil-lations appear to be suppressed when a B-spline sub-basis is applied with a number of functions a # n sothat no multiple nodes occur; see Fig. 1~b! with a 5 5.Third, in general a distribution grows steeper on theleft-hand side as on the right-hand side. Therefore Ipropose to deal with a nonequidistant node grid byusing the roots of the Chebyshev polynomials on theleft-hand side, indicated by asterisks in Fig. 1~a!.
The right choice of suitable B-spline functions isessential. An anticipated shape of the solution func-tion x strongly suggests a particular choice of a basis.
Fig. 1. ~a! B-spline basis of the order of k 5 4 on a nonequidistantgrid, ~b! favorable subbasis of the hybrid regularization technique.
20 March 2001 y Vol. 40, No. 9 y APPLIED OPTICS 1333
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h1sobcprpus
wN
0
rmal
1
It should also be observed that in many cases thecondition number increases with k, the order of the Bsplines. Ill-conditioning can lead to a solution xathat provides unsatisfactory solutions. On the onehand the magnitude of the condition number, in it-self, does not determine the success or the failure ofthe Galerkin or collocation method. For example, ifone is fortunate enough to choose a basis set thatcontains a function approximately equal to the truesolution of the given equation, then one can antici-pate good results for small a, and no conditioningproblem; see Fig. 2~a!. On the other hand, it is ob-viously desirable to minimize the overall error by anadroit choice of a, the regularization by control ofdimensionality, as stated in Subsection 3.A; see Fig.2~b!, which is still a little better than Fig. 2~a!.
Higher-order B-splines are advocated for the basisunctions ~see Fig. 2! as a better choice than the piece-ise constant or piecewise linear basis functions that
Fig. 2. Reconstruction results of the seventh example in Table 2with 5 1 2 noiseless input data, i.e., the fourth setup case in Table1, by means of different numbers a of B splines and orders k: ~a!a 5 4, k 5 4, g 5 0; ~b! a 5 9, k 5 6, g 5 0; ~c! a 5 8, k 5 8, g 5; ~d! a 5 10, k 5 9, g 5 0.
Table 2. Different Simulation Examples of Monomodal Logarithmic-NoSetup Ca
Example 1 2
Radius rmod ~mm! 0.05 0.1Derivation s 1.8 1.4Refractive index m 1.7 1 0.01i 1.4 1 0.5iRadius reff ~mm!, exact 0.12 0.13
Retrieving error of reff ~%!1. Setup case 6 1 2 0.16 0.132. Setup case 6 1 0 5.12 0.855. Setup case 3 1 1 14.23 1.39
Regularization parameters: a, k, g1. Setup case 6 1 2 7, 4, 0 17, 5, 02. Setup case 6 1 0 8, 4, 0 15, 5, 15. Setup case 3 1 1 4, 4, 0 7, 4, 0
334 APPLIED OPTICS y Vol. 40, No. 9 y 20 March 2001
ave been suggested in the methods given in Refs. 18,9, and 43. The successful application in Fig. 2hows that the inversion process is not sensitive torders k between 4 and 9 as well as to numbers aetween 4 and 10. However, with respect to theondition number of matrix ~17! the interval @4,5# isroposed for the second integer regularization pa-ameter k. The interval of the first regularizationarameter a depends on the number of wavelengthssed and on the noise level. An interval @3,25# isuggested; see Tables 2 and 3.
C. Regularization by means of Truncated Singular-ValueDecomposition
To stabilize the numerical algorithm, an additionalregularization technique is proposed, e.g., TR orTSVD, in the second stage. Many aerosol applica-tions use the Twomey–TR solution44 x 5 ~ATA 1gH!21ATy with a real regularization parameter g,where the matrix H is nearly diagonal and dependson the smoothing constraint of choice. Often, as inthe algorithm in Ref. 19, the second-difference ex-pression is chosen as a measure of smoothness, sothat the smoothest solution is one that minimizesthis expression. Although the minimization of thesecond-difference expression has become the com-monly used smoothness criterion, it may not alwaysbe the most appropriate.13 For example, if a solu-tion has several sharp peaks, e.g., bimodal or multi-modal size distributions, these might be altogethereliminated by the second-difference criterion.
For this reason we use TSVD regularization. Lin-ear system ~16! now has the form
(i51
a
*rmin
rmax
k~lj,r!Nik~r!dr big 5 y~lj! 1 dj,
j 5 1, . . . , m 5 N 1 M, (20)
hich may be underdeterminate or overdeterminate.and M are explained in Table 1. We solve Eq. ~20!
Distributions and the Inversion Results of Noiseless Input Data, Threef Table 1
Table 3. Different Simulation Examples of Bimodal Distributions ~Top! and Gamma Distributions ~Bottom! and the Inversion Results of Noiseless
by TSVD. The SVD of A is a decomposition of theform
A 5 UDm,aVT 5 (
i51
min~m,a!
niuiviT, (21)
where U 5 ~u1, . . . , um! and V 5 ~v1, . . . , va! arematrices with orthonormal columns and the blockdiagonal matrix Dm,a 5 diag@n1, . . . , nmin~m,a!# has
onnegative diagonal elements appearing in nonin-reasing order such that n1 $ n2 $ . . . $ nmin~m,a!
$ 0.he numbers ni . 0 are the singular values of A,
and the vectors ui and vi are the left and the rightsingular vectors of A, respectively.
In connection with discrete ill-posed problems, twocharacteristic features of the SVD are often found.First, the singular values decay gradually to zero withno particular gap in the spectra. An increase of thedimension of A will increase the number of small sin-gular values. Second, the left and the right singularvectors tend to have more sign changes in their ele-ments as the index i increases, i.e., ni decreases. Bothfeatures24,40 are consequences of the fact that the SVDof A is closely related to the SVE of the underlyingoperator K in Eq. ~6!. The third integer regulariza-tion parameter g [ @0, min~m, a! 2 1# is the truncationlevel of the singular values. That parameter cuts offthe singular values in order to eliminate round-off ef-fects and to smooth out the solution. It holds
bg 5 ~A†!g~y 1 d! 5 V~Da,m†!gUT~y 1 d!, (22)
Input Data, the S
Examples
Radii rmod1, rmod2
~mm! 0.05,Derivations s1, s2 1.8, 2Refractive index m 1.55 1
reffexact~mm! @retrieving error ~%!# 0.58 ~0
atexact~mm2cm23! @retrieving error ~%!# 5.47 ~0
vtexact~mm3cm23! @retrieving error ~%!# 1.06 ~0
ntexact~cm23! @retrieving error ~%!# 51 ~2
Regularization parameters: a, k, g1. Setup case 6 1 2 12, 4,
p1, p2 5.3 3p3, p4 8.94,Refractive index m 1.4 1
reffexact~mm! @retrieving error ~%!# 0.89 ~1
atexact~mm2cm23! @retrieving error ~%!# 164 ~0
vtexact~mm3cm23! @retrieving error ~%!# 49 ~1
ntexact~cm23! @retrieving error ~%!# 100 ~5
Regularization parameters: a, k, g1. Setup case 6 1 2 8, 4, 1
ith
~Da,m†!g 5 diagS 1
n1, . . . ,
1nmin~m,a!2g
, 0, . . . , 0D ,
(23)
1nj
: 5 51nj
nj . 0
0 nj 5 0, (24)
j 5 1, . . . , min~m, a! 2 g. The solution xa,k,gd can be
represented as xa,k,gd~r! 5 ¥i51
a bigNik~r!.
4. Numerical Results
Equations ~1! and ~2! are formulated into a morepecific form,
y~lj! 5 *rmin
rmax
Kv~r, l; m!v~r!dr, (25)
ith
˜ v~r, lj; m! :5 HKpv~r, lj; m! lj [ Ll
p
Kextv~r, lj; m! lj [ Ll
ext , (26)
where y~lj! are the optical data, either backscatter bor extinction a, depending on lj. Different commonsetup cases are specified in Table 1. The v~r! term isthe volume concentration distribution. It holds
20 March 2001 y Vol. 40, No. 9 y APPLIED OPTICS 1335
r
w
4
1
A. Retrieval of the Particle Distribution
Simulated inversions have been made for five differ-ent setup cases. The reconstruction results of theinversion from synthetic optical data were comparedwith the input distribution. Monomodal and bi-modal logarithmic-normal distributions and gammadistributions were used to describe the particle sizedistribution function. We deal with seven examplesof the monomodal distribution ~see Table 2!,
n~r! 51r
1
Î2p ln sexpF20.5
~ln r 2 ln rmod!2
ln2 s G ,
(28)and with three examples of the bimodal and the mod-ified gamma distribution ~see Table 3!,
n~r! 5 (j51
2 1r
1
Î2p ln sj
expF20.5~ln r 2 ln rmod j
!2
ln2sjG ,
(29)
n~r! 5 p1 rp2 exp~2p3r p4!, (30)
espectively. Here, rmodjare the mode radii, sj are
the mode widths, p1 is the total concentration, and p2,p3, and p4 are positive constants. In addition, thegamma distribution45 is a common one used to de-scribe water cloud droplet and dust distributions.
The values for the refractive indices in Tables 2 and3 correspond to typical values found in practice.46–49
The range of typical refractive indices lies between1.33 and 1.8 for the real part mR, e.g., mR 5 1.4corresponds to sea salt particles, mR 5 1.55 to conti-nental particles, and mR 5 1.7 to particles of burnprocesses, and between 0 and 0.2 for the imaginarypart mI. The value mI 5 0.5 corresponds to a specialsituation for testing the stability of the algorithmbecause the degree of ill-posedness40 of the operatorof Eq. ~25! becomes higher if the imaginary part in-creases. To choose ranges for the mode radius andwidth, it must be observed that only particles withradii of the order of the magnitude of the measure-ment wavelengths possess particle-size-dependentscattering efficiencies2,14,20 and therefore are suitablefor inversion. Thus the range of detectable particlesize distributions is limited. Therefore mode radiibetween 0.05 and 0.3 mm and mode widths between1.4 and 2.0 were chosen; see Tables 2 and 3. Thesenumbers correspond approximately to the accumula-tion mode45 that essentially covers those particlesthat cause radiative forcing in the atmosphere andthus have an effect on climate.
The present method uses two steps to find the so-lution, i.e., the hybrid inversion for each regulariza-tion triples by recalculation of the optical data. Theselection of the solution takes place by use of themaximum norm with respect to the difference of theinput and the reconstructed optical data, but is re-jected if the nonnegative constraint is not fulfilled.In cases in which the order of magnitude between theextinction and the backscatter data is extremely dif-ferent, we use an additional weighting.
336 APPLIED OPTICS y Vol. 40, No. 9 y 20 March 2001
First, the results of the monomodal distributionsare presented. The inversion results of noiseless op-tical data are excellent as the reconstructed distribu-tion ~dotted curves in subsequent figures! is more orless equal to the input distribution ~solid curves insubsequent figures! for different setup cases in Table1; see Fig. 2 ~fourth lidar setup case!, Fig. 3~a! ~firstsetup case!, Fig. 3~c! ~third setup case!, and Fig. 4~a!
Fig. 3. Reconstruction results of the volume distribution fornoiseless data ~example 7 in Table 2! with ~a! 6 1 2 ~first setup casein Table 1! and ~c! 5 1 2 ~third setup case in Table 1! wavelengthsand for noisy data ~15%! with ~b! 6 1 2 and ~d! 5 1 2 wavelengths.The regularization parameters are ~a! a 5 12, k 5 4, g 5 0; ~b! a 54, k 5 5, g 5 1; ~c! a 5 7, k 5 5, g 5 0, ~d! a 5 3, k 5 5, g 5 0.
Fig. 4. Reconstruction results of the volume distribution fornoiseless data ~example 7 in Table 2! with ~a! 6 1 0 ~second setupcase in Table 1! and ~c! 3 1 1 ~fifth setup case in Table 1! wave-lengths and for noisy data ~10%! with ~b! 6 1 0 and ~d! 3 1 1
avelengths. The regularization parameters are ~a! a 5 4, k 5 5,g 5 1; ~b! a 5 3, k 5 4, g 5 0; ~c! a 5 6, k 5 5, g 5 0, ~d! a 5 3, k 5, g 5 0.
atbmobb
wGdtr3
~second setup case!, Fig. 4~c! ~fifth setup case!, Figs.5~a!–5~f ! ~first setup case!, and Figs. 6~a!–6~f ! ~fifthsetup case!. In contrast to the order k of B splines,the regularization parameter a strongly depends onthe number of wavelengths and the noise level. Inthe noiseless data case the parameter a varies be-tween 7 and 17 in the first setup case in Table 1,whereas in the second and the fifth setup cases theparameter a lies between 4 and 16 or 4 and 8, respec-tively; see Table 2. This means that the smaller thenumber of wavelengths, the smaller the dimension ofthe projection space must be. From a mathematicalpoint of view, that is clear as the available informa-tion of the system becomes smaller. Moreover, weobserve that the third regularization parameter g isneeded in only a third of the examples and even thereis cutting only one singular value off necessary. Insummary, in almost all cases the two regularizationstages work so well hand in hand that even the firststage, i.e., the regularization by discretization, hasalready completely fulfilled the regularization pro-cess or the second stage has to complete only a smallremainder of work; see Table 2.
Second, we prove gamma distributions; see Table 3~bottom! and Figs. 7~b!, 7~d!, and 7~f !. Although weobserve the first setup case in Table 1 ~the parametera varies between only 5 and 8, in contrast to thelogarithmic-normal distributions!, the obtained re-
Fig. 5. Example ~a! 1 of Table 2, ~b! 2, ~c! 3, ~d! 4, ~e! 5, ~f ! 6, allwith noiseless input data as well as with 6 1 2 backscatter andextinction coefficients, i.e., the first setup case in Table 1.
sults are just as good as those for the previous dis-tribution. In addition, the gamma distribution is anexcellent example to show that the algorithm doesnot require any a priori knowledge of the analyticalshape of the distribution. The inversion results inFig. 7 show that it is even possible to recover anydistribution shape.
Third, we examine bimodal logarithmic-normaldistributions; see Table 3 ~top! and Figs. 7~a!, 7~c!,nd 7~e!. The algorithm detects bimodal distribu-ions as well. The errors in the determined distri-utions are slightly larger compared with those foronomodal distributions. Whereas the choice k 5 5
r k 5 4 is again a good one, the parameter a had toe increased between 7 and 22. That is explainableecause now two peaks had to be recovered.Finally, the stability of the inversion with noisy dataas tested by the addition of statistical errors with aaussian distribution to the correct synthetic opticalata. Standard deviations from 5% to 20% were usedo simulate erroneous optical data. If we test incor-ect optical data with, for example, 15% noise @Figs.~b! and 3~d!# or 10% noise @Figs. 4~b! and 4~d!#, we
observe logically that the first regularization parame-ter a must again be smaller than in the noiseless cases,because the second summand of relation ~15! on theright-hand side would otherwise become larger, i.e.,the algorithm must smooth out the noise. Figures3~b! and 3~d! show that this reconstruction method
Fig. 6. Example ~a! 1 of Table 2, ~b! 2, ~c! 3, ~d! 4, ~e! 5, ~f ! 6, allwith noiseless input data as well as with 3 1 1 backscatter andextinction coefficients, i.e., the fifth setup in Table 1.
20 March 2001 y Vol. 40, No. 9 y APPLIED OPTICS 1337
vinmt
1
provides excellent results up to 15% noise in the firstand the third setup cases and fairly good results up to10% noise in the second and the fifth setup-cases; seeFigs. 4~b! and 4~d!. It should be pointed out that thereconstructions of Figs. 3~b!, 3~d!, 4~b!, and 4~d! aretypical for the above-mentioned noise realization. Ingeneral, our experiences show that the best choice innoisy data cases seems to be almost always a 5 4 ora 5 3.
B. Simultaneous Retrieval of Distribution and RefractiveIndex
Besides the volume distribution, the complex refrac-tive index is a second unknown quantity during theinversion process. This problem is a highly nonlin-ear one and, in general, no explicit information on therefractive index is available. Although we knowthat the refractive index depends on the wavelengthand sometimes also on the particle size, we assumethe index to be constant.
By using a refractive-index grid between suitablelimits mRmin
, mRmaxand mImin
, mImaxwith a sufficiently
small step size, we solve the inversion problem of thelast subsection for each grid point and try to find aunique minimum with respect to the maximum normin the valley. If the valley is too flat or if the solutiondomain is disconnected, we try to enclose a solution
Fig. 7. ~a! Example 1 of Table 3 ~top!, ~b! example 1 of Table 3~bottom!, ~c! example 2 of Table 3 ~top!, ~d! example 2 of Table 3~bottom!, ~e! example 3 of Table 3 ~top!, ~f ! example 3 of Table 3~bottom!, all with noiseless input data as well as with 6 1 2 back-scatter and extinction coefficients, i.e., the first setup case in Table 1.
338 APPLIED OPTICS y Vol. 40, No. 9 y 20 March 2001
domain of possible refractive indices. Actually, weobtain excellent results with that technique; see Figs.8–11. The resulting range with an arrow or withtwo arrows is the dark area inside the purely whitearea with respect to a suitable noise level, dependingon the noise level of the optical data.
In contrast to the fact that the degree of ill-posedness40 is higher with respect to the inverseproblem with known refractive index if the absorp-tion is strong, here we observe the opposite. If mI 50.5 @see Figs. 10~a! and 11~a!#, the solution domaincan easily be defined, which is plausible. If aslightly incorrect refractive index is used, the inverseproblem responds with a large output error that isdue to the high ill-posedness degree. Although thereal part is always slightly overestimated, the deter-mination of the imaginary part is excellent. Theweak absorbing cases are not quite so stable; see Figs.8~a! and 9~a!. We observe that the solution domainis more elongated and often disconnected. In con-trast to the previous strong absorption case here thereal part mR is well included in the domain.
Figures 8~b!–11~b! show the excellently recoveredolume distributions in the appropriate refractive-ndex minimum points. We have to point out that inoisy data cases, not shown here, the minimum pointust not be the best solution. Rather, all points of
he dark area are to be regarded as suitable solutions.
Fig. 8. Example 7 in Table 2 ~but with mexact 5 1.5 1 0.0i! and thefirst setup case in Table 1 with an additional unknown refractiveindex: ~a! reconstruction of a refractive-index range with an errorsmaller than 1.0% with respect to the backscatter and extinctioncoefficients, ~b! reconstruction of the volume distribution by use ofthe determined refractive index m 5 1.51 1 0.0025i; the regular-ization triple was determined to a 5 6, k 5 5, and g 5 0.
sltFftais
Tc
d
C. Retrieval of Microphysical Parameters
The mean and the integral properties of the particleensemble that were calculated from the inverted par-ticle distribution were chosen as effective radius, i.e.,surface-area weighted mean radius
reff 5* n~r!r3dr
* n~r!r2dr
, (31)
as total surface-area concentration,
at54p * n~r!r2dr, (32)
as total volume concentration,45
vt54p
3 * n~r!r3dr (33)
and as number concentration of particles,
nt5* n~r!dr. (34)
Fig. 9. Example 7 in Table 2 and the first setup case in Table 1with an additional unknown refractive index: ~a! reconstructionof a refractive-index range with an error smaller than 1.2% withrespect to the backscatter and extinction coefficients, ~b! recon-struction of the volume distribution by use of the determined re-fractive index m 5 1.5 1 0.01i; the regularization triple was
etermined to a 5 12, k 5 4, and g 5 0.
First, we observe that in the first setup case withnoiseless data the retrieval of the effective radius, thesurface-area, and the volume concentration is notcritical for the monomodal logarithmic-normal casesas well as for the cases of bimodal and gamma dis-tributions. It is possible to get results with a rela-tive error well below 3%; see Tables 2–5. Thenumber concentration is a more sensitive value.Whereas in the monomodal case it is possible toachieve results below 10% or 15% for the logarithmic-normal or the gamma distribution, respectively, inthe bimodal case one can expect errors as high as50%; see Tables 2–5.
Second, we examine the second setup case. Be-cause from a mathematical point of view the degree ofill-posedness40 of integral equation ~1! is smaller thanthe degree of Eq. ~2!, another realistic way is to ob-erve the retrieval process when only a backscatteridar setup without any Raman channel is used. Inhis way it is possible to attain suitable results; seeig. 4~a!. Otherwise the retrieval errors of the ef-
ective radius, surface area, and volume concentra-ion become larger compared with the first setup casend the retrieval of the number concentration is crit-cal; see Table 4. Therefore the two Raman channelstabilize the retrieval process.Third, we investigate the minimal fifth setup case.
he reconstruction of the volume distribution is suc-essful; see Figs. 4~c! and 6. Although the errors
Fig. 10. Example 2 in Table 2 and the first setup case in Table 1with an additional unknown refractive index: ~a! reconstructionof a refractive index range with an error smaller than 1% withrespect to the backscatter and extinction coefficients, ~b! recon-struction of the volume distribution by use of the determined re-fractive index m 5 1.438 1 0.49i; the regularization triple wasdetermined to a 5 18, k 5 5, and g 5 1.
20 March 2001 y Vol. 40, No. 9 y APPLIED OPTICS 1339
wn
o
bttpwpbcift
ttet
r
Table 5. Relative Errors ~%! of the Inversion of Mean Microphysical
fi
1
~see Table 4! increase, the first three parameters stayell below 7%, and it is not astonishing that theumber concentration is again critical.Finally, we deal with noisy data. Each measure-
ment error simulation and its inversion has beenrepeated ten times to obtain more general results.We observe the same effect for the number concen-tration; see Table 5, last column. Roughly speaking,for input noise on the optical data of as much as 10%or 20%, the retrieval of the effective radius, surfacearea, and volume concentration with a relative errorat approximately 5% or 10%, respectively, is possible.Additionally, we observe that the retrieval becomesmore ill-posed40 if the imaginary part of the refractiveindex increases. Therefore the relative errors of theretrievals of the effective radius and of the volumeconcentration become visibly larger if the imaginary
Fig. 11. Example 2 in Table 2 ~but with mexact 5 1.5 1 0.5i! andhe first setup case in Table 1 with an additional unknown refrac-ive index: ~a! reconstruction of a refractive index range with anrror smaller than 1.14% with respect to the backscatter and ex-inction coefficients, ~b! reconstruction of the volume distribution
by use of the determined refractive index m 5 1.52 1 0.5i; theegularization triple was determined to a 5 15, k 5 4, and g 5 0.
Table 4. Relative Errors ~%! of the Inversion Results of MicrophysicalParameters from Noiseless Input Data for the First Six Examples inTable 2, each Example with Six Different Refractive Indices, and for
340 APPLIED OPTICS y Vol. 40, No. 9 y 20 March 2001
part is larger than 0.1i; see Table 5. This effectccurs only if the input noise is less than 15%.Instabilities in the inversion process cannot always
e completely suppressed. In the first line, however,hese instabilities influence only the shape of the par-icle distribution. It could be shown that the meanroperties in these cases can also be reproduced fairlyell. In summary, it follows that not only can the sizearameters of a particle distribution be determined,ut also information on the complex refractive indexan be gained. Furthermore it could be shown that,n the first setup case, errors of 15–20% are tolerableor stable inversion results of microphysical parame-ers.
5. Conclusions
The inversion method with hybrid regularization hasbeen chosen for routine calculation of physical parti-cle properties from lidar measurements of backscat-ter and extinction coefficients. The regularizedretrieval of the volume distribution with the espe-cially designed hybrid technique was outstandinglysuccessful. Finally, one needs only a lidar setupwith three backscatter and one independent extinc-tion wavelengths of a Raman channel as a minimalconfiguration. The combination of backscatter andextinction information is not strictly necessary, but isrecommended for the inversion process because theinversion errors increase slightly without extinctioninformation. In detail, with the extensive setup, i.e.,six backscatter and two extinction wavelengths, themean and the integral parameters of the particle sizedistribution, except the number concentration, can belimited to a 3% error or to a 10% error for the noise-
Parameters of Monomodal Logarithmic-Normal Distributions withNoiseless Input Data for Examples 1, 2, and 4–6 with 18 Different
Refractive Indices and with Noisy Data for Example 5 with 5 DifferentRefractive Indicesa
aTen runs were used for each level of random noise and sepa-rated into two groups with the imaginary part of the refractiveindex mI smaller or larger than 0.1i. All computations use the
rst setup case of Table 1.
hflnlpsmorttw
less data or for the 20% noisy data, respectively.Without any extinction wavelengths, i.e., only sixbackscatter wavelengths, the parameters can be lim-ited to a 5% error as well as for the minimal setupcase to well below 10% when noiseless data are used.
This method can be used to retrieve all shapes of dis-tribution, especially multimodal ones. Additionally, asuccessful technique was proposed to capture the solu-tion domain of the unknown refractive index, which is ahighly nonlinear problem. The algorithm has alreadysuccessfully inverted experimental data sets2 within theframework of an aerosol closure experiment, the Linden-berg Aerosol Characterization Experiment ~LACE98 inGermany, July–August 1998!.
At the moment, the algorithm works with a suc-cessful but heuristic choice of the optimal regulariza-tion parameters by means of forward calculationswith the best orders of k between 4 and 5, with thebest parameters of a between 3 and 25, and TSVD.There are three important possibilities of numeri-cally more efficient choices. First, the discrepancyprinciple is a simple computational approach withthe advantage of being time efficient. However, ithas the drawback that it oversmooths the solution.13
Second, the general cross-validation ~GCV! ap-proach26 that is used, e.g., in the method of Ref. 19,
as the drawback that the GCV function can have aat minimum and hence may be difficult to locateumerically. Moreover, GCV cannot handle corre-
ated measurement errors. The L-curve method is aopular one. The method strikes a balance betweenmoothness and fidelity to measurements. Further-ore, the method yields results that are independent
f experimental errors and can therefore handle cor-elated errors. A drawback of the L-curve method ishat it is not convergent for highly multimodal dis-ributions. The research to improve our algorithmith the L-curve method50 to find simultaneously all
three optimal regularization parameters more effi-ciently is ongoing.
Currently, only cases of homogeneous sphericalparticles are considered by our inversion method.All nonspherical and inhomogeneous effects are twoother sources of error. Some of those have alreadybeen examined in Refs. 51 and 52. To describe theseerrors, models are necessary that permit one to cal-culate the backscatter and extinction behavior andthe difference with respect to spherical particles.Respective algorithms have been developed.53,54
The study of these influences is ongoing.Finally, one can deal with descriptive regulariza-
tion techniques by using objective a priori informa-tion about the solution such as the nonnegativeconstraint of the size distribution. Unfortunately,such constraints55,56 do not form a compact set andtherefore do not lead to a well-posed problem. How-ever, conjugate gradient projection57 allows one toutilize various kinds of additional information aboutthe function shape sought, in particular their piece-wise monotonicity and piecewise convexity. Thisutilization of a priori shape constraints imposed onthe solution and having stabilizing properties pro-
duces a much stronger regularization effect than thetraditional smoothness conditions and ensures, at thesame time, preservation of main qualitative charac-teristics of the sought for solution. Other usefultools are neural networks58,59 and maximum entropyregularization.32 These possibilities might be incor-porated into future versions of our algorithm.
The author thanks Detlef Muller of the Institutefor Tropospheric Research in Leipzig, Germany, forhis suggestions of the test examples. The author isgrateful to the anonymous referees for their valuablecomments and suggestions, in light of which this pa-per has been revised. This study has been sup-ported by the Bundesministerium fur Bildung,Wissenschaft, Forschung and Technologie undergrant 07AF310y2.
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